The crossover between weak-coupling BCS theory of superconductivity and strong-coupling BEC describes the fundamentals of quantum bound states of particles, including information on the superconducting critical temperature (\(T_c\) ) and the possible pseudogap formation [1]}, [2]}. In the strong-coupling regime, the pair-formation temperature \(T_{\rm pair}\)  and the actual superconducting transition temperature \(T_{\rm c}\)  are distinctly separated (Fig.REF (a)). Between \(T_{\rm pair}\)  and \(T_{\rm c}\) , the so-called preformed Cooper pairs exist, which give rise to large superconducting fluctuations and possibly a depletion of the low-energy density of states (DOS), namely the pseudogap. Almost all experimental studies on the crossover physics have been performed in ultracold atomic systems for the past decades [2]}, because of the difficulty in tuning the attractive interaction between electrons in solids. Therefore, it is extremely challenging to realize such strongly-coupled pairs in electron systems. In underdoped high-\(T_{\rm c}\)  cuprates, the pseudogap formation has been discussed in terms of the BCS-BEC crossover [4]}, but recent experiments support a phase transition of different kinds at the onset of pseudogap and the relevance of BCS-BEC crossover in cuprates remains unresolved [5]}.
